Quick pointer, I found this one slightly harder to write & wrap my head around so any comments guiding me towards a more accurate conclusion would be appreciated!

What is Bayesian Updating?

“A statistics professor who travels a lot was concerned about the possibility of a bomb onboard his plane. He determined the probability of this and found it to be low, but not low enough. So now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being onboard would be infinitesimal.”

This is a classic joke among mathematicians to explain Bayesian Updating. In short, the theory of Bayesian Updating concerns how we should adjust probabilities when we encounter new data.

It takes it’s roots from an essay published in the 18th century by Thomas Bayes, an English minister. The essay, called “An Essay toward Solving a Problem is the Doctrine of Chances”, was only surfaced two years after the minister’s death but gave birth to the theory.

That theory rests on a basic premise, namely: take the odds of an event happening & adjust for new information. If you have a strong prior knowledge about an event then the probability you produce after Bayesian Updating will have a higher chance of accuracy.

Think of it like this:Initial Beliefs + Recent Objective Data = A New and Improved Belief

Although Bayesian Updating, or Bayesian “Inference”, is now commonly used in statistical engineering, you don’t necessarily have to take a maths based approach to situations to get the most out of this mental model.

Using Bayesian Updating in sports betting

If you do want to take a methodical approach to Bayesian Updating when making smart decisions, consider the following put forward by Pierre-Simon Laplace who was a French mathematician & astronomer in order to boil Bayes’ theorem into a formula:

P (A/B) = P(B/A) x P(A) / P(B)

If you want to know the probability of A when you know that B is also present (given), you can get the answer by multiplying your prior estimation of A (Probability of A) by how much more likely B is when A is present (i.e. P(B|A)/P(B)).

Let’s take a simple football match betting example to start. Let’s say it’s Chelsea v Man Utd. Chelsea have an overall head-to-head win percentage of 31% against Man Utd. We also know that when Cheslea win against Utd it rains 11% of the time, compared to the usual likelihood of rain in a Chelsea Match of 10%. So:

P(A) = probability that Chelsea beats Man Utd = 31%

P(B) = Probability of rain in a Chelsea match = 10%

P(A|B) = Probability of rain in a football game when Chelsea beats Utd = 11%

Let’s now imagine it’s game day & the weather forecast is rain. A simple Bayesian update will show you that P(A|B)=P(A)*P(B|A)/P(B)= 31%*11%/10%= 34.1%.

So Chelsea have a 34.1% chance of winning.

Using Bayesian Updating in DFS

You don’t have to go through all this when playing DFS. All that’s important is understanding the underlying logic that when new information is presented, you should update your existing assumptions.

It can be as simple as a last minute formation change by a manager from 4-4-2 to 3-5-2 & asking yourself if the midfielder you’ve initially drafted will be as effective in this formation.

However, Bayesian Updating does have it’s limitations. Just because the volume of information increases every day, doesn’t make the future any less predictable.

Shane Parrish recalls the example of a turkey who is fed every day. Every day that he’s fed he thinks that in general, life is a breeze where he’s fed by friendly humans every day. Externally we know this is because he’s being prepared for slaughter.

A recap

The availability heuristic – the inability to extract information past the last event that occurred, leading to a biased view of performance.

The bias from liking – an inherent & natural bias in all of us to place undue value on something or someone simply because we like it.

The gambler’s fallacy – a misunderstanding of the concept of probability.

Time to develop the gambler’s fallacy into it’s more sophisticated cousin, the law of the regression to the mean.

What is “Regression to the mean”?

When Eden Hazard scored 3 goals against Newcastle in 2014, the tendency is either to: think he outworked himself & is going to crash in the next game or that he’ll score another hat-trick. The reality is that in the next game he’ll still play well, just likely not 3-goals well. He’ll regress back to his mean performance.

This is the concept of regression to the mean, that in any situation where there’s many variables & chance is involved, extreme outcomes tend to be followed by more moderate ones. This shouldn’t be confused with the fallacious law of averages, where an unusual amount of successes (or wins, goals etc) should be balanced by losses.

Rather, it’s that good teams or good players will continue on average to perform well, just not as well as their last exceptional performance which was most likely due to an extra bit of luck.

Let’s put it another way. The above chart plots the correlation between weight & height, clearly showing a weak to moderate correlation. What this means is that while height is generally a good predictor of weight, there are lots of other factors at hand that push results above & below the mean. Therefore, when the correlation between two things is less than perfect then we must be wary of regression to the mean.

Whenever the correlation between two scores is imperfect, there will be regression to the mean.

Regression to the mean in DFS

Clearly then, this concept should be of use to DFS players looking to extract value out of DFS games with a salary cap / budget functionality. If an expensive player like Hazard performs consistently well during the season, then it’s no use trying to be overly contrarian & not drafting him because he had a suspiciously good last game.

Rather, you should understand that regularly drafting Hazard over the course of a season is likely to bring you on average a higher return on your investment than other players, because the “mean” that his scores regress to is higher than other players.

Another important point to understand is that just because an unusual player is top projected right now, or has scored 5 times in a row against the next opposition he’s playing against, doesn’t mean he will stay top or score again. There is good fortune involved, so when their performances regress to the mean they’ll be overtaken.

If I could sum up the key takeaway it’s this: remember the importance of track records rather than one-time success stories. When you look at the graphic below of Eden Hazard’s track record in the Premier League the stats don’t lie. Out of 174 EPL games, his goals per game stands at 0.33. Even if next game he scores 5 goals, that’s hardly going to move that average significantly up or down. That’s the beauty of sample sizes.

Another perfect example of this is Leicester winning the Premier League, which many argue arose from a combination of chance factors (not discrediting their achievement here.) Rating Leicester as likely of winning the league again going forward simply isn’t sustainable.

Don’t be fooled by one time events, always respect the regression to the mean.

The Story So Far

So far we have explored how the “Availability Heuristic” tricks us into remembering events that are most easily available to our memory, and how the “Liking Bias” clouds our judgement by putting our favourite players ahead of statistical logic. This week we’ll explore The Gambler’s Fallacy, otherwise known as “The Monte Carlo” Fallacy.

Why?

Because although DFS is a game of skill, a poorly trained mind that falls trap to the biases that usually befall gamblers then guess what, you’ll end up gambling. Guessing, making rash decisions, panicking & losing isn’t what DFS is about.

What is The Gambler’s Fallacy?

Think of a roulette table in Vegas. The board about the table shows the last 15 spins in a row hitting red. How many times have you thought, “it’s bound to hit black next”? The sequence of events has clouded your judgement, because in your mind it’s increased the likelihood of change.

This is a misunderstanding of the concept of probability. If the two colours on the table (forget 0 for now) are red & black, the probability of either is always 50%. The same with a coin flip.

If you wanted to place a definition on it, you could say it’s “the flawed reasoning that, in a situation of pure random chance, the outcome can be affected by previous outcomes.”

Let’s think about it another way. Would you ever choose the numbers 1, 2, 3, 4, 5, 6 on your lottery ticket? Most people would think that’s stupid. It’s too ordered, a more random sequence like 6, 14, 22, 35, 38, 40 is more likely to come up. However, the order from 1 to 6 is made up in our imagination. We’re the ones that give it any meaning. Random chance is order & sequence agnostic.

The Gambler’s Fallacy in DFS

I don’t want to discredit the importance of a team or player’s form when making decisions in DFS. Clearly past performance can be a good indicator of how well a player will turn up next game. That’s why top players will show consistency in their last handful of games.

As with anything in life however, balance is key. Understanding how probability works & the role of chance in any sporting situations will make you a better DFS player. If you treat each game as an independent trial, you’ll be less likely to allow biases to fool you.

The fact that Giovinco has only scored once in his last five MLS games is not an indication that he’s “due” to score. Similarly, if a defensive player like Nemanja Matic has scored in his last three games, this “hot streak” is likely nothing more than a product of variance.

There are certain players who are more likely to score than others. Sergio Ageuro, Diego Costa, Lionel Messi. However, those players are usually out of most DFS player’s budgets (assuming you’re playing a game that uses salary caps.) So how do you make sure you’re still making smart decisions while trying to find value in lesser known players?

The first step is avoiding gambler’s fallacy. It’s essentially the easiest route to take because you simply look at players on hot streaks.

Recap

Last week I wrote about how the availability heuristic impairs us as daily fantasy players from making smart decisions. We remember more clearly those events which are more readily available to our memory, which is especially prevalent in daily sporting contests.

This week I’ll move to look at the bias from liking, which sounds a little more obvious, but tricks all of us all the time. It’s slightly different from the availability heuristic in that it’s not necessarily what we last remember, but subconsciously how we’re wired to think.

What is bias from liking?

Have you seen MoneyBall? There’s a scene where the scouts are sitting there considering replacements for their lost MVP, and are chatting away about how one candidate isn’t the right one. The reason? He has an “ugly girlfriend”, which must mean a lack of self confidence.

Billy Beane sits there with his head in his hands, “what are we doing?” He’s already realised the importance of removing our biases & focusing on the data. The bias in this particular situation? Bias from liking. Bias from liking good looking players, players who just look right, speak right & have good looking girlfriends.

We’ve all fallen victim to this “Halo Effect”, where we automatically ascribe certain characteristics to individuals when we have no idea whether they have them, simply because of how they look. A good looking woman is probably kind & a tall handsome man is probably smart. It’s all a sub-conscious system working away.

This isn’t a bias confined to the individual either. In the US & Canada, attractive individuals earn on average 12-14% higher than their unattractive coworkers. It’s a systematic subconscious, but it can be beaten.

“And what will a man naturally come to like and love,

apart from his parent, spouse and child?

Well, he will like and love being liked and loved.”— Charlie Munger

Bias from Liking in DFS

Think about a full EPL weekend, where all the players are on show. You have a full set of players to choose from, your sample size is as big as can be. What’s your first reaction? Probably the usual names that you like, the players from your own team.

That’s why FPL restrict the number of players you can choose from one team, you’d end up picking them all. That’s why most DFS sites have a budget, so you can’t just go in & pick your favourite players every time. You have to think.

Even with these restrictions however, you’re still at risk. Let’s imagine a situation where Chelsea are playing Man Utd, and Liverpool are playing West Brom. You’re making your striker decision. First thought? Zlatan Ibrahimovic. Why? You’re a Man Utd fan, you love Ibra!

You probably feel guilty for even thinking about drafting a Liverpool striker. “Scum”, you think to yourself. Last week I mentioned the idea of “the matrix of fact,” which is a term I’ve borrowed from my legal studies in the past. It was set out by the famous Lord Hoffman in a 1998 contract case. The basic premise is that all the facts of situation, both direct & indirect, should be taken into account before determining the context of that situation.

In the above example, you have to acknowledge that Man Utd’s opposition are Chelsea here. The forward players aren’t likely to get many chances, and once you add the variable of DFS scoring into the equation, you realise you have to forget about who you like & think which forward will be most effective on this DFS site today.

It is clear then that Firmino for Liverpool will likely get more touches, key passes, shots, assists and goals against West Brom than Ibrahimovic against Chelsea. You might like Ibra more as a player, but when you take into consideration his price & estimated value add against Chelsea, it’s better for you to go against your gut feeling.

What are “mental models?”

The first time I came across the concept of using mental models in decision making was when listening to AngelList founder, Naval Ravikant, on “The Knowledge Project” podcast by Shane Parrish. The way he explained how any problem ranging from maths to philosophy to predictions, can be solved or avoided using a “lattice framework” of mental models has stuck with me ever since.

“A lousy way way to do…prediction” he says, “is X happened in the past, therefore X will happen in the future. What you want is principles. You want mental models.”

Why do we need mental models?

We all know that DFS, every day, comes with a lot of tough decisions. Many, like SaahilSud, use statistical models & analytical tools to get the upper hand. He’s even launched his own lineup builder, RotoQL, in which DK CEO Jason Robins invested in (a questionable move by many in the community). But for those of us who want to play more casually, yet still have a good chance of winning, what mental models can we use off the top of our head to ensure we make smart decisions every time? I feel like most casual players will enter a 1v1 draft or small contest thinking “Eden Hazard scored the last time I watched him play, he’s bound to score again.” Lousy.

I’ll be posting an article a week of the best mental models when it comes to making decisions in daily fantasy football. In some games, like DraftKings & FanDuel where huge GPPs mean smart decisions must be made at every turn, these mental models can be applied to existing mathematical frameworks to ensure higher chances of success. In others, like DRAFT or Dribble, they can be used as standalone to give confidence in making smart decisions when it comes to drafting.

What is the “availability heuristic?”

Let’s imagine that you’re walking down a busy street by a river on a Saturday. You see a busy restaurant filled with people. It’s brimming, you turn to your friend and say “I think I’ll invest in the riverside restaurant business, it seems booming!”

You’ve made two mistakes here. First, you’ve ignored the matrix of fact and second, you’ve failed to take into account the importance of base rates. What you’ve done fantastically well, is base your potential decision on the information immediately available to you. You’ve looked at one restaurant out of thousands in that area and decided that, because this one is full on this particular day, that the riverside restaurant business is worth investing in.

If you considered the matrix of facts you’d realise that of course the restaurant is busy on this particular Saturday. It’s a hot day, so anything by the river is attractive for foot-fall. It’s also a Saturday, so customers are more naturally looking for a place to eat. However, visit the same spot every day for the rest of the year & you’ll understand why it’s so important to consider all the facts before making assumptions. How’s business when it’s raining & cold?

I know I said that mental models are supposed to be quick frameworks to avoid spending ages on statistical analysis, but base rates are such a simple thing to figure out. A simple Google search shows you 60% of restaurants don’t make it past the first year & 80% go under in five. Now you’re thinking, is there anything about this particular restaurant that could pull it away from the base rate & into the 20% who survive past 5 years? That’s a better way to think.

The availability heuristic at work in DFS

Back to DFS, I experience this all the time. The most available information to me is the last game I watched, or the last set of highlights on YouTube. This is dangerous because we misjudge the frequency & magnitude of recent events. If I don’t know much about the Premier League & I see Michael Carrick score cracking goal, I think a great goal scorer. We also have limitations on memory, so it’s important to look at the matrix of facts when it comes to reviewing the fixtures & which teams are playing each other.

In “Thinking Fast and Slow”, Kahneman writes:

“People tend to assess the relative importance of issues by the ease with which they are retrieved from memory—and this is largely determined by the extent of coverage in the media.”

Articles & news reports about upcoming games are important to take into account into the matrix of fact, but don’t let them be your guiding hand. The writers themselves are often influenced by what they last saw. Don’t let your emotions get in the way either, especially in soccer, where passions can run high. Emotions in soccer shapes our intuitive perceptions and tricks us.

Beat the bias

So how can you use the availability heuristic to your advantage in DFS to gain the upper hand? Well, if the bias is inherent in all of us, it’s inherent in your opponents too. It’s therefore profitable to be contrarian, to look at players who haven’t necessarily done well in their last match but have a high average in their last five.